To solve the equation [tex]\(2^{16 + 8x} = 5^{10 + 8x}\)[/tex], let's break it down step-by-step.
1. Express in terms of logarithms:
We start by taking the logarithm of both sides to utilize the properties of logarithms which can help us isolate [tex]\(x\)[/tex]. Let's use the natural logarithm ([tex]\(\ln\)[/tex]) for convenience:
[tex]\[
\ln\left(2^{16 + 8x}\right) = \ln\left(5^{10 + 8x}\right)
\][/tex]
2. Apply the power rule of logarithms:
The power rule states that [tex]\(\ln(a^b) = b \cdot \ln(a)\)[/tex]. Applying this property, we get:
[tex]\[
(16 + 8x) \cdot \ln(2) = (10 + 8x) \cdot \ln(5)
\][/tex]
3. Expand both sides:
Distribute the logarithms:
[tex]\[
16 \cdot \ln(2) + 8x \cdot \ln(2) = 10 \cdot \ln(5) + 8x \cdot \ln(5)
\][/tex]
4. Isolate the terms involving [tex]\(x\)[/tex]:
To isolate [tex]\(x\)[/tex], let's get all the terms involving [tex]\(x\)[/tex] on one side and the constant terms on the other:
[tex]\[
8x \cdot \ln(2) - 8x \cdot \ln(5) = 10 \cdot \ln(5) - 16 \cdot \ln(2)
\][/tex]
5. Factor out [tex]\(x\)[/tex]:
Combine the [tex]\(x\)[/tex]-terms:
[tex]\[
x \cdot (8\ln(2) - 8\ln(5)) = 10\ln(5) - 16\ln(2)
\][/tex]
6. Simplify the coefficient of [tex]\(x\)[/tex]:
Notice the common factor of 8 in the terms involving [tex]\(x\)[/tex]:
[tex]\[
x \cdot 8 (\ln(2) - \ln(5)) = 10 \cdot \ln(5) - 16 \cdot \ln(2)
\][/tex]
7. Solve for [tex]\(x\)[/tex]:
Divide both sides by the coefficient of [tex]\(x\)[/tex]:
[tex]\[
x = \frac{10 \cdot \ln(5) - 16 \cdot \ln(2)}{8 (\ln(2) - \ln(5))}
\][/tex]
Given the standard logarithmic properties, this equation represents the algebraic manipulation needed to isolate and solve for [tex]\(x\)[/tex]. The exact form can involve complex expressions if we simplify further, but this captures the exact reasoning path. The numerical or simplified solution from here is tricky without a calculator, but that's the condensed, principled approach to solving exponential equations logarithmically.
The solution, as achieved, can be rendered in more compact or numeric formats based on specific computational tools. Mathematical tools or software can further refine this solution's format into more familiar logarithmic or decimal representations directly.
